The quantization of the two-dimensional toric and spherical phase spaces is considered in analytic coherent state representations. Every pure quantum state admits there a finite multiplicative parametrization by the zeros of its Husimi function. For eigenstates of quantized systems, this description explicitly reflects the nature of the underlying classical dynamics: in the semiclassical regime, the distribution of the zeros in the phase space becomes one-dimensional for integrable systems, and highly spread out (conceivably uniform) for chaotic systems. This multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems.